The three- and four-loop equal-mass sunset integrals and local Calabi--Yau fourfolds and fivefolds
Abstract
We determine the equal-mass two-dimensional three- and four-loop sunset Feynman integrals I(3)circleddash and I(4)circleddash at large Euclidean momentum in terms of the genus-zero Gromov--Witten theory of the local Calabi--Yau fourfold and fivefold XL=Tot(KFL), where FL⊂(ℙ1)L+1, L=3,4, is a smooth hypersurface of multidegree (1,…,1). At two loops the sunset integral is governed by the prepotential of a local Calabi--Yau threefold. For L≥3 the moduli spaces of unmarked stable maps have positive virtual dimension and there is no prepotential, therefore the Feynman integral is governed instead by the one-point potential ℱγL with insertion γL=HL−1/(L−1)!, by its gravitational descendants, and by ζ(3) boundary terms coming from the Gamma class. At both loop orders the coefficient of RL−1 in the Frobenius expansion in the mirror coordinate R is −θQℱγL with the Novikov variable Q=exp(R). At three loops the whole Feynman integral is an explicit expression in ℱγ3 alone, because genus-zero recursion determines all descendant series from it; at four loops two diagonal series suffice, and the ζ(3) term is itself corrected by θQℱγ4. At three loops we also derive the analytic normalization independently, from the Picard--Fuchs equation and the polar part of a Mellin transform, prove integrality of the multiple-cover transforms for arbitrary integral insertions together with an explicit quantum-K formula, and compute the exact period connection polynomials. Finally we give endpoint values along the Euclidean axis, bounds on the Taylor radii in the Novikov variable Q, a computer-assisted proof of the four-loop endpoint nonanalyticity, and a comparison of the normalizations at two, three and four loops.
Related
- Equal-mass sunset integrals and genus-zero local Gromov--Witten theory of Calabi--Yau n-folds
- The Tate conjecture for abelian fivefolds over finite fields
- The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds
- Irrational families of Fano weighted complete intersection fourfolds
- Ambient Chow Rings of Debarre--Voisin Fourfolds and the Beauville--Voisin Conjecture
BibTeX
@misc{maennel2026three,
title = {{The three- and four-loop equal-mass sunset integrals and local Calabi--Yau fourfolds and fivefolds}},
author = {Hartmut Maennel and Pierre Vanhove},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10821},
eprint = {2610.10821},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05788327v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The three- and four-loop equal-mass sunset integrals and local Calabi--Yau fourfolds and fivefolds
AU - Hartmut Maennel
AU - Pierre Vanhove
PY - 2026
DA - 2026/10/07
PB - HAL
DO - 10.48550/arxiv.2610.10821
UR - https://hal.science/hal-05788327v1
AB - We determine the equal-mass two-dimensional three- and four-loop sunset Feynman integrals I^(3)_circleddash and I^(4)_circleddash at large Euclidean momentum in terms of the genus-zero Gromov--Witten theory of the local Calabi--Yau fourfold and fivefold X_L=Tot(K_F_L), where F_L⊂(ℙ^1)^L+1, L=3,4, is a smooth hypersurface of multidegree (1,…,1). At two loops the sunset integral is governed by the prepotential of a local Calabi--Yau threefold. For L≥3 the moduli spaces of unmarked stable maps have positive virtual dimension and there is no prepotential, therefore the Feynman integral is governed instead by the one-point potential ℱ_γ_L with insertion γ_L=H^L−1/(L−1)!, by its gravitational descendants, and by ζ(3) boundary terms coming from the Gamma class. At both loop orders the coefficient of R^L−1 in the Frobenius expansion in the mirror coordinate R is −θ_Qℱ_γ_L with the Novikov variable Q=exp(R). At three loops the whole Feynman integral is an explicit expression in ℱ_γ_3 alone, because genus-zero recursion determines all descendant series from it; at four loops two diagonal series suffice, and the ζ(3) term is itself corrected by θ_Qℱ_γ_4. At three loops we also derive the analytic normalization independently, from the Picard--Fuchs equation and the polar part of a Mellin transform, prove integrality of the multiple-cover transforms for arbitrary integral insertions together with an explicit quantum-K formula, and compute the exact period connection polynomials. Finally we give endpoint values along the Euclidean axis, bounds on the Taylor radii in the Novikov variable Q, a computer-assisted proof of the four-loop endpoint nonanalyticity, and a comparison of the normalizations at two, three and four loops.
ER -